This paper is concerned with complementing the known analytical studies of the pure coalescence equation. There is still a need for better analytical tools for the analysis of this problem even though high-speed computers have contributed much to the knowledge of this system. Specifically, when the detailed microphysics is incorporated into a large-scale, three-dimensional, moist, deep-convection model, it is currently impossible to solve the coalescence equation numerically for several size categories. Hence, there is a need for better analytical tools. In particular, we are concerned with the relationships between integral power moments of the size spectrum and the collection kernel, relationships between Friedlander's similarity solutions and the kernel, bounds for the size spectrum, and various power-moment inequalities. The results we obtained will allow us to make reasonable approximations for spectra which can, in turn, be used in large-scale convection models.